t31os' answer is helpful, but it only goes as deep as the container
template, so if I call his function in header.php, it will tell me
page.php instead of header.php. In other words, I want to know the
exact file my function is calling out from.

This is a similar issue as the first question, and still using t310s'
answer as reference.

Also looked into the wp_get_theme function, but that object only
gives me info about the theme itself, rather than an array or list of
files.

My goal is to write a function that would dynamically echo out all the file names or pieces on what's being viewed on the front-end . So a "Page" would echo page.php, sidebar.php, footer.php and header.php for example.

Please no plugins, I'd like to resolve this with PHP. I get that this might not be a simple answer, but I'd like a starting point at the very least - or at least know if it's possible.

Of course you can use in header.php too, but from that file you'll get only the files loaded at the moment header.php has been included, e.g. page.php and header.php. PHP is a programming language, not a magic machine, and can't know which files are going to be included before they are actually included.

I like this solution because I don't have to use multiple functions to grab the header/footer/sidebar and then the general templates. It's all consolidated into one class and feels much cleaner. I've tried it out and I am getting all the templates printed in the footer. Of course I can re-work it so it's not printing out the array of paths - but this is a great start! Thanks so much!
– RachieVeeNov 12 '14 at 14:41

get_header, get_footer, and get_sidebar all have corresponding actions that pass the name as an argument, so you can hook those actions, test for their existence via locate_template and log the names.

get_template_part has an action, but it's dynamic and includes the slug, so unless you know the slugs, those will be difficult to hook. what you can do is hook all, and string match the first part of the action, get_template_part_, which is exactly what this What the File plugin does:

The locate_template function is only used in a conditional to find a file I specify, correct? And then if I have it hook into get_header, I have to write a function to display "header.php" on the front-end if locate_template for header.php returns true? Just trying to confirm I'm thinking about this right. Thanks. :-)
– RachieVeeNov 11 '14 at 21:55

yes, you're using locate_template strictly to see if templates exist. you don't have to do anything on the get_ hooks except test and log, those functions just trigger the actions and continue doing their thing on their own.
– MiloNov 11 '14 at 22:02